Physics of magnetically dominated plasma: dynamics, dissipation and particle acceleration

1. Physics of magnetically dominated plasma: dynamics, dissipation and particle acceleration Maxim Lyutikov (UBC) Cracow, June 2006

2. Dynamics and energy dissipation in electromagnetically-dominatedplasmas Maxim Lyutikov (McGill University, CITA) Cracow, June 2003

3. Relativistic outflows may be produced and collimated by large scale B-fields • AGN jets: ergosphere and Black hole itself can act as a Faradey disk (Blandford-Znajek, Lovelace), creating B-field dominated jets • Numerical simulations begin to show this dynamically (Koide et al. 2002), (McKinney, Gammie, Krolik, Proga) Large scale, energetically dominant magnetic fields may be expected in the launching region of relativistic jets and may (should?) continue into emission regions

4. In plasma rest frame: Magnetic energy U’B=B2/8π, Plasma energy: rest-mass, ρc2 Alfven 4-velocity Magnetization parameter (σ1/2= μ1/3) Magnetically dominated: σ > 1 σ > Γ2/2 σ ~ 1 (Subsonic relativistic flow) σ ~ 0 σ = ∞ III. Force-free (magnetodynamics) I. Hydrodynamics (all fluid) II. MHD Non-fluid Fluid shocks (e.g. Fermi) Magnetic dissipation New plasma physics regime: magnetically dominated plasma Dissipation & acceleration

5. Dynamics

6. supersonic (MHD), Γ2> σ, in vacuum: flow acceleration determined by internal flow dynamics: flow passes through fast sonic points (eg ΓF~√σ), becomes causally disconnected (Michel) subsonic, Γ2< σ, : acceleration limited by external medium, causally connected flows pressure balance at the contact Γ r Expansion of σ >> 1 wind

7. σ > Γ2/2– subsonic flow σ < Γ2/2– supersonic flow (reached terminal Γ0) At late times, t> > tGRB (self-similar), composition of ejecta is not important At early times: MHD, σ < Γ2/2 Force-free σ > Γ2/2 σ > 1: weak or no reverse shock emission tobs~tGRB Γ=Γ0 σ < Γ2/2 GRBs: σ < Γ2/2 and σ > Γ2/2 have different early dynamics σ →∞ subsonic Γ ~ t -1/2 Γ Γ~ t-3/2 supersonic Self-similar Sedov-Blandford-McKee tcoord (Lyutikov 2003,2006)

8. steep shallow flares t~102-103 s steep t~104-5 s (Nousek; Zhang) Observations? • Swift results are very puzzling: flares and lighcurve breaks at t ~10 -- 105 sec (two “breaks” were expected, tGRB~100 sec and @ Γ~1/θ, 105 sec) • For σ < 1 strong reverse shock emission is expected • For σ >1 no reverse shock is weak or non-existent • Reverse shock in fireball: same type as internal shocks: microphysics is fixed by prompt emission • Expected optical flash m ~ 12-18. • Cooling: Flux ~ t -2 ; later: cooling to radio emission. • In the Swift era absolute majority of GRBs do not show predicted RS behavior (despite UVOT and numerous robotic telescopes). • This may indicate highly magnetized ejecta, σ > 1 • Other possibilities to produce some optical emission (e.g. e± by γ)

9. Long GRBs: expansion inside a star of a σ> 1 wind. As long as expansion is non-relativistic there must be dissipation • Energy and Bφ-flux is injected linearly with v~c • for non-relativistic expansion volume is near constant , Bφ ~ t • Energy ~ Bφ2 ~ t2 ??? (c.f. Gunn & Rees PWNs) • Need to destroy Bφ – flux: inductance break down → dissipation • Energy goes into e±-γ (~ first 3 sec): lost after photosphere • This is different from AGNs (c.f magnetic tower of Lynden-Bell), where expansion can always be relativistic, but not for GRBs (Lyutikov & Blandford, 03)

10. Dissipation in magnetically-dominated plasma

11. ω ωB ωp Cascade k Dissipation: σ > 1 – energy in B-field • σ > 1: shock are weak; do not exist for σ> σcrit • B-field dissipation due to current instabilities (“reconnection”) • B-fields are strongly non-linear systems: dissipation property of the emission region, NOT of the source activity (e.g. Solar B-field generated on ~22yr time scales, flares can rise in minutes) • σ > 1 – new plasma regime • Adopt non-relativistic schemes: • Magnetodynamical tearing mode • Relativistic reconnection • new acceleration schemes (no hydro or non-relativistic analogues) • Charge-starved plasma, turbulent EM cascade 3-wave processes are allowed: FFA, AAF! (in non-relativistic MHD 3-wave with ω ≠ 0 are prohibited)

12. Resistive instability of relativistic force-free current layer(unsteady reconnection) • Resistivity is usually very small (τR~ L2/η >> τ) • Current sheets are unstable – formation of small scale sub-sheets • Tearing mode τ ~ (τAτR)1/2 τA ~ L/vA ~L/c, τR~ L2/η • Similar to hydro (waves forms shocks) resistive RFF forms dissipative current layers • Essential for RFF simulations, EM turbulent cascade (ML,03)

13. X X j j }δ Dissipation rate ~ ηΔB~ηB/δ2 B Electric field B E=ηII jII Tearing mode in σ=∞ plasma • σ= ∞ : matter inertia is not important, force-free currents ensure JxB + ρe E=0 and decay resistively j║= ρp v║,p - ρe v║,e~ E|| /η j┴=( ρp - ρe )v┴ parallel currents attract formation of current sheet (Lyutikov 03) - very fast Resistive (tearing) EM instability New plasma physics regime, same expression for growth rate? (come from very different dynamical equations: Maxwell and MHD)

14. slow motion in σ = ∞ plasma Non-linear stage: formation of magnetic islands (Komissarov et al, 2006) Tearing mode in σ=∞ plasma very similar to incompressible MHD! Growth rates in excellent agreement with analytics This may be a step towards formation of reconnection layers. Applications: magnetars (growth rate ~ msec, similar to flare rize time), AGN, GRB jets

15. Giant flare SGR 1806: Time scales: observed rize time, < 250 μsec, implies reconnection in the magnetosphere (Alfven time , t ~ RNS/c ~ 30 μsec) Similar to Solar Coronal Mass Ejection (CME). Magnetar jets (plumes)? Late constant velocity, sub-relativistic outflow may be just a projection effect θ Applications: magnetars, AGN, GRB jets. (Lyutikov 2006) (Palmer et al. 2005) βapp= β ctg θ/2

16. Acceleration of UHECRs

17. UHECRs: • Emax ~ 3 1020 eV • Isotropic, perhaps small scale clustering • UHECRs must be produced locally , < 100 Mpc • Perhaps dominated by protons above ~ 1018 eV • Hard(ish) aceleration spectrum, p ~ 2-2.3

18. Acceleration by large scale inductive E-fields: E~∫ v•E ds • Potential difference is between different flux surface (pole-equator) • In MHD plasma is moving along V=ExB/B2 – cannot cross field lines • Bring flux surfaces together –Z-pinch collapse (Trubnikov etal95) • Kinetic motion across B-fields- particle drift - (Bell, Blasi, Arons) E V B Ud

19. Lovelace 76 Blandford 99 B E ┴ B : Inductive potential E Pictor A (FRII) To reach Φ=3 1020 eV, LEM > 1046 erg/s (for protons) This limits acceleration cites to high power AGNs (FRII, FSRQ, high power BL Lac, and GRBs) There may be few systems with enough potential within GZK sphere (internal jet power higher than emitted), the problem is acceleration scheme

20. V V E~x E~-x B B Potential Φ~-x2 ∆Φ~√L/c Potential Φ~x2 Potential energy of a charge in a sheared flow E=-v x B Depending on sign of (scalar) quantity (B *curl v) one sign of charge is at potential maximum Protons are at maximum for negative shear (B *curl v) < 0

21. V B v E x Bφ Er Er Astrophysical location: AGN jets • There are large scale B-fields in AGN jets • Jet launching and collimation (Blandford-Znajek, Lovelace) • Observational evidence of helical fields • Jets may collimate to cylindrical surfaces (Heyvaerts & Norman) • Jets are sheared (fast spine, slow edge) Protons are at maximum for negative shear (B *curl v) < 0. Related to (Ω*B) on black hole Ω BH and disk can act as Faradey disk

22. B-field F Udrfit ~ F x B ∆ B ud F ∆ BφxBφ Er Drift due to sheared Alfven wave • Electric field Er ~- vz x Bφ : particle need to move radially, but cannot do it freely (Bφ). • Kinetic drift due to waves propagating along jet axis ω=VA kz • Bφ(z) → Ud ~ ~er

23. Why this is all can be relevant? Very fast energy gain : • highest energy particles are accelerated most efficiently!!! • low Z particles are accelerated most efficiently!!! (highest rigidity are accelerated most efficiently) • Acceleration efficiency does reach absolute theoretical maximum 1/ωB • Jet needs to be ~ cylindrically collimated; for spherical expansion adiabatic losses dominate

24. Acceleration rate DOES reach absolute theoretical maximum ~ γ/ωB • Final orbits (strong shear), rL ~ Rj, drift approximation is no longer valid • New acceleration mechanism • For η < ηcrit < 0, ηcrit = - ½ ωB/γ, particle motion is unstable • non-relativistic: • Acceleration DOES reach theoretical maximum ~ γ/ωB • Note: becoming unconfined is GOOD for acceleration (contrary to shock acceleration) η=V’

25. ankle γ Mixed composition protons Spectrum • From injection dn/dγ~γ-p →dn/dγ ~ γ-2 Particles below the ankle do not gain enough energy to get rL ~Rj and do not leave the jet UHECRs are dominated by protons, below the ankle: Fe

26. Astrophysical viability • Need powerful AGN FR I/II (weak FR I , starbursts are excluded) • UHECRs (if protons) are not accelerated by our Galaxy, Cen A or M87 • Several powerful AGN within 100 Mpc, far way → clear GZK cut-off should be observed: Pierre Auger: powerful AGNs? • GZK cut-off • few sources • IGM B-field is not well known • Fluxes: LUHECR ~ 1043 erg/sec/(100 Mpc)3 – 1 AGN is enough

27. Gradient of Rotation Measure across the jet Gradient of liner polarization B (Gabuzda 03, confirmed by Taylor etal) Faradey Rotation and gradient of linear polarization across the jet in 3C 273 Possible interpretation: helical field Need lots of poloidal flux → may come from a disk, not BH

28. Conclusion • EM-dominated plasma: may be a viable model for a variety of astrophysical phenomena. Very little is done. • Macrophysical models (ideal dynamics) • Microphysics (resistivity is anomalous, η~c2/ωp, η~c2/ωB; particle acceleration) • Need for simulations (both dynamics and acceleration) • EM codes with currents • PIC codes (Nordlund) • Observations seem to be coming along

30. “Where have you seen plasma, especially in magnetic field?” Landau • “The magnetic field invoked is proportional to someone’s ignorance” Woltijer

31. Radiative losses • Equate energy gain in E =B to radiative loss ~ UBγ2 • As long as expansion is relativistic, total potential remains nearly constant, one can wait yrs – Myrs to accelerate

32. Two parameters: Lundquist S=VAL/ η »1, σ » 1 outflow is always relativistic Inflow: σ « S – non-relativistic inflow S « σ « S2 – relativistic inflow Relativistic reconnection can be fast, ~ light crossing time Relativisticreconnection, σ>> 1(Sweet-Parker) (Blackman & Field 1994; Lyutikov&Uzdensky2003; Lyubarsky 2004)

33. Particle acceleration in relativistic reconnection Leptons • Numerical experiments are only starting (Hoshino02, Larrabee etal 02). • Spectra depend on kinetic properties and geometry ∫(v•E)dl (McClements) • If escape ~ rL, then • For GRB we need γ-1 (Lazatti), also TeV AGNs (Aharonian) • No calculations of acceleration at relativistic tearing mode (should accelerate as well)

34. X X Dissipation rate ~ ηΔB~ηB/δ2 }δ j j B E=η j B Why magnetic energy wants to dissipate What is needed for magnetic dissipation is presence of electrical current parallel currents attract formation of current sheet Anomalous resistivity η(j) Electric field Next: generalize non-relativistic fluid models to new regime

35. Wave surfing can help • Shear Alfven waves have δE~(VA/c) δB, • Axial drift in δExB helps to keep particle in phase • Particle also gains energy in δE δE B ∆ γmax/γ0 ud Alfven wave with shear Er Most of the energy gain is in sheared E-field (not E-field of the wave, c.f. wave surfing) Alfven wave without shear

36. AGN jet • In situ acceleration is required (tsynch< R/c, short time scale variability: 20 min at TeV!) • e± winds - strong losses at the source • Ion-dominated - hard to get variability, low radiation efficiency (Celotti,Ghisellini) EM- dominated! • Currents needed for collimation; Currents are unstable • Resistive modes may not destroy the jet, but re-arrange it (eg, sawtooth in TOKAMAKs, Appl) • Relativistic FF jets stabilized by rotation • Hard power law may be needed for TeV emission(Aharonian) • Polarization from helical B-field (Gabuzda; ML, in prep) (Lesch&Birk;Lovelace;ML)

37. Jets start as B-field-dominated, can σ changes on the way? (Weber & Davis, Goldreich & Julian Vlahakis &Konigl) • Ideal conversion: acceleration • Acceleration to fast point Γfast ~ √σfast • At this point σ ~ Γ2>> 1 : flow remains B-field dominated • Collimation σ→ 1, but it is slow ~ ln z and unlikely σ << 1 • There are some indication (Homan et al , Jorgstad et al.) moving features, (Sudou et al.) increased jet-counter jet brightness, but not conclusive (jet bending & aberration can give visible acceleration). • Dissipative: on scale > RBHΓ2 ~ 10 17 cm (e.g relativistic reconnection βin ~ 1, Lyutikov & Uzdensky) • blazar γ-ray emission zone (Lyutikov 2003). Variation in Γ produced locally (no large UV variations of disk are seen): (Sikora et al. 2005) • Jet can remain B-field dominated to pc scales

38. Shocks Spectra of Fermi-accelerated particles (kinetic property) can be derived from shock jump conditions Electrons need to be pre-accelerated to γ~ mp/me ~ 2000 (or √mp/me ~43) B-field “Reconnection” spectra are not “universal”, depend on details of geometry (universal in relativistic case, p=1 ?) No need for pre-acceleration: all particles may be accelerated How can the two paradigms (σ>>1 and << 1) be distinguished? 1. Acceleration scheme with predictive power (γmin, p)

39. Shock typically produce p>2, relativistic shocks have p ~ 2.2 (Ostrowki; Kirk) non-linear shocks & drift acceleration may give p<2, e.g. p=1.5 (Jokipi, Bell & Lucek) B-field dissipation can give p=1 (Hoshino; Larrabee et al.); such hard spectra may be needed for TeV emitting electrons (γ-γ pair production on extragalactic light Aharonyan; Schroedter). How can the two paradigms be distinguished?: very hard spectra, p< 2 p< 2 spectra should not be discarded as unphysical

40. γ Γ bulk Γ ~ 2 Γbulkγ 2. Radiation modeling: not conclusive Blazar dominance by IC: Uph’ /UB’ ~ Γ4, Γ > 10; U e± ’ < UB’ < Uph’ • TeV : SSC, B-field strongly under equipartition @ 1017 cm → Γ~50-100 (Krawczynski 04). Outflows in bulk flow? • Equipartition (e.g in FR II hotspots, Hardcastle) • U’B ~ U’ e± : • Amplification: sub-equipartition • Dissipation → equipartition: (μe=γ me c 2/B)

41. In plasma frame In laboratory frame B' e' n' e'┴ B', n' Aberration of Π: B-field is NOT orthogonal to polarization Both B-field and velocity field are important for Π (Blandford & Konigl 79; Lyutikov,Pariev,Blandford 03)

42. B Π from relativistically moving cylindrical shell with helical B-field Γ=2, ψ’=π/4, θob=π/3 B not orthogonal to e Jet can be Bφ dominated in observer frame and Bz-dominated in rest

43. l v shock Π from random B-field compressed at an oblique shock Π l upstream shock downstream Π not aligned with projection of l, also Cawthorn & Cobbs Lyutikov (in prep)

44. Aberration of Π • Direction of Π depends both on B-field and velocity field • Always plot “e”, not “inferred” B-field • One needs to know velocity to infer internal B-field • Symmetries in the velocity field may help • e.g. if shock is conical, on average polarization along or across jets (Cawthorn & Cobbs)

45. Π from cylindrical shell with helical B-field Π depends on p Even co-spatial populations with different p may give different Π (eg Radio & Optical) Π along the jet Π across the jet Γ=10, p=1, different rest frame pitch angles

46. Bimodial distribution of PA PA follows the jet as it bends Sometimes a bend gives 90 change of PA For cylindrical jet U=0, average Π along or across the axis. Only conical shocks can give the same. For fixed ψ, Π mostly keeps its sign. Note: for plane shock there is no correlation between Π and bend direction. Sometimes a change does occur Large scale or small scale B-fields in pc-scale AGNs jets (Aller et al) BL Lac 1749+701 (Gabuzda 03)

47. (Gabuzda 03) e e Resolved jets • Resolved jets: center: PA ║, edges: PA ┴ • Emission is generated in small range Δr < r (shear acceleration? Ostrowski) • Core is boosted away Limb-brightening Mkn 501, (Giroletti )

48. Jet polarization may tell the spin of BH Right Left Left & Right helixes look different Different Π signature Direction of BH or disk spin (if θΓ is known)

49. Cen A (Hardcastle et al 2003) Radio X-ray Tests/unresolved issues • Firmly established flow acceleration at (sub)-pc scales: evidence of B-field conversion • More Π studies, especially CP (unidirectional B-field) • with MHD codes • Spectra requiring p → 1 • Acceleration rates above • Very high Π > 50% in R • compressed B-fields isotropize on Aflven time scale, 3-D random B-field is dominated by small scale • turbulence will lead to isotropization • Different e-acceleration mechanisms? • X-rays are displaces from O-R (e.g. Cen A) • Magnetic & shock accleration? (Kirk) • NB: similar in Crab pulsar

50. Are all ultra-relativistic jet the same? • Epeak – L correlations • GRBs - positive, BL Lac – negative • Internal shocks in GRBs must be highly (unreasonably?) efficient, in BL Lac – inefficient • GRS 1915: jets appear after drop of the x-ray flux, blazars: no correlation between UV flux and flares • jets without BH (Cirnicus X-1)